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Full text of "Analytical Mechanics"

MOTION OF A PARTICLE                       113
But if TI and r2 denote the distances of the particles from the axis of rotation, and P the period of revolution, then
Therefore                         -1 = ~2
ra2     r-i
gives the ratio of the masses of the particles as well as those of the given bodies.
MOTION OF A PARTICLE UNDER A CONSTANT FORCE.
102. Case I. Rectilinear Motion.  Suppose a particle of mass m to be acted upon by a force F, which is constant in direction as well as in magnitude. Then the force equation gives
dv     F     ,                              /TA
or                                     -T; = ~ = /.                          (I7)
dt     m                                  K  '
Since both m and F are constant, /, the acceleration, is also constant. Integrating equation (I') once we obtain
v =ft+c,
where c is a constant to be determined by the initial conditions of the motion. Let the initial velocity be denoted by vQj then v= vQj when =0, therefore c= VQ and
v = v* + ft.                                 (1)
Substituting -    for v in equation (1) and integrating,
Let   s = 0, when t = 0 ; then c! = 0.    Therefore
8 =^+!/*2.                                (2)
Eliminating t between equations (1) and (2) we get
v*=vo* + 2f8.                                 (3)